In many applications voltage reference circuits operate under strongly changing temperature conditions.
U.S. Pat. No. 3,887,863 discloses to controllably operate two transistors at markedly different emitter current densities for deriving a temperature-independent reference voltage. A control loop may be used to force the collector currents of the two transistors to be equal. The two transistors may have different sizes of emitter areas. A first resistor connecting the emitter of a first of both transistors to ground of a DC power supply may be used to generate a voltage across the first resistor which may be proportional to absolute temperature (PTAT).
As described in Tsividis, Y.: “Accurate Analysis of Temperature Effects in IC-Vbe Characteristics with Application to Bandgap Reference Sources”, IEEE journal of solid-state circuits, vol. sc-15, no 6, December 1980, page 1078-1084, the base emitter voltage Vbe of a transistor, in particular a bipolar transistor, may exhibit a dependence on the absolute temperature T which can be described with the mathematical formula (Equation 1):
            Vbe              Q        ⁢                                  ⁢        1              =                  V                  G          ⁢                                          ⁢          0                ′            -                                                  V                              G                ⁢                                                                  ⁢                0                            ′                        -                          Vbe                              Q                ⁢                                                                  ⁢                                  1                  R                                                                          T            R                          ·        T            -                        V          T                ·                  (                      n            -                          x              1                                )                ·                  ln          ⁡                      (                          T                              T                R                                      )                                ,
where:
V′G0 represents a bandgap voltage of a semiconductor material, extrapolated to 0 degrees Kelvin; the semiconductor material may be silicon;
VbeR represents a base-emitter voltage at temperature TR;
VT=kT/e represents a thermodynamic voltage, wherein k represents the Boltzmann constant, and e represents the electron charge;
T represents an absolute temperature in Kelvin;
TR represents a reference temperature in Kelvin;
n represents a process-dependent parameter; n represents a temperature-independent parameter; n may be 4 minus the power of a temperature dependency of an (effective) mobility for minority carriers;
and
x1 may represent a power of temperature dependency of the collector current of the first transistor under operating conditions. x1 may depend on the bias current; it may, e.g., be 1 if the bias current is proportional to absolute temperature or may be 0 when the current is temperature-independent.
As can be seen from the term VNL=−VT (n−x1)ln(T/TR) in Equation 1, the base-emitter voltage Vbe(T) may exhibit a non-linear dependency over temperature T. This term may change the output voltage of a conventional Brokaw cell in an undesired manner. Usually, the factor (n−x1) cannot be set to zero to compensate for the non-linear term.
Thomas H. Lee: “Handout #20: EE214 Fall 2002: Voltage References and Biasing”, rev. Nov. 27, 2002 (available at www.stanford.edu/class/archive/ee/ee214/ee214.1032/Handouts/ho20bg.pdf) discloses that the parameter n is typically a minimum of 2 and range up to about 6. Usually, the parameter n may be close to 4. Typical values of (n−x1) may range from 1 to 5, and usually may be close to 3. Even if the value of (n−x1) was 1, the term VNL=(n−x1) VT ln (T/TR) would be still non-linear and would still be not zero. The temperature drift of a conventional Brokaw cell caused by the non-linear term VNL is typically not higher than 1% of the output voltage VOUT.